.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/curves/plot_02_hopf_umlaufsatz.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_curves_plot_02_hopf_umlaufsatz.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_topology_curves_plot_02_hopf_umlaufsatz.py:


Hopf's Umlaufsatz: the turning number of a closed curve (1935)
==============================================================

As you walk once around a closed plane curve, your heading turns by a
whole number of full turns. Hopf proved that for every *simple* closed
curve that number is :math:`\pm 1`. Curves that cross themselves can
have any turning number: 0 for a figure eight, 2 for a limaçon with an
inner loop, 4 for a circle carrying three small inner loops.

.. GENERATED FROM PYTHON SOURCE LINES 13-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import turning_number

    t = np.linspace(0, 2 * np.pi, 800, endpoint=False)
    rng = np.random.default_rng(3)
    radius = 1 + sum(rng.normal(0, 0.12) * np.cos(k * t + rng.uniform(0, 6)) for k in range(2, 7))
    curves = {
        "ellipse": np.column_stack([2 * np.cos(t), np.sin(t)]),
        "random blob": radius[:, None] * np.column_stack([np.cos(t), np.sin(t)]),
        "blob, clockwise": (radius[:, None] * np.column_stack([np.cos(t), np.sin(t)]))[::-1],
        "figure eight": np.column_stack([np.sin(t), np.sin(2 * t)]),
        "limaçon": (0.5 + np.cos(t))[:, None] * np.column_stack([np.cos(t), np.sin(t)]),
        "three inner loops": np.column_stack([np.cos(t) + 0.6 * np.cos(4 * t), np.sin(t) + 0.6 * np.sin(4 * t)]),
    }

    fig, axes = plt.subplots(2, 3, figsize=(11, 7))
    for ax, (name, c) in zip(axes.ravel(), curves.items(), strict=True):
        ax.plot(*np.vstack([c, c[:1]]).T, lw=1.5)
        # arrows show the direction of travel
        for i in range(0, len(c), len(c) // 6):
            d = c[(i + 3) % len(c)] - c[i]
            ax.annotate("", c[i] + d, c[i], arrowprops={"arrowstyle": "->", "color": "tab:red"})
        ax.set_title(f"{name}\nturning number {turning_number(c)}")
        ax.set_aspect("equal")
        ax.axis("off")
    fig.suptitle("Hopf (1935): a simple closed curve turns exactly once")




.. image-sg:: /api/gallery/topology/curves/images/sphx_glr_plot_02_hopf_umlaufsatz_001.png
   :alt: Hopf (1935): a simple closed curve turns exactly once, ellipse turning number 1, random blob turning number 1, blob, clockwise turning number -1, figure eight turning number 0, limaçon turning number 2, three inner loops turning number 4
   :srcset: /api/gallery/topology/curves/images/sphx_glr_plot_02_hopf_umlaufsatz_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 0.98, 'Hopf (1935): a simple closed curve turns exactly once')



.. GENERATED FROM PYTHON SOURCE LINES 43-47

The tangent angle along the curve
---------------------------------

The unwrapped heading rises by :math:`2\pi` per turn.

.. GENERATED FROM PYTHON SOURCE LINES 47-56

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(8, 4))
    for name in ("random blob", "figure eight", "limaçon", "three inner loops"):
        c = curves[name]
        d = np.roll(c, -1, axis=0) - c
        ax.plot(t, np.unwrap(np.arctan2(d[:, 1], d[:, 0])) / (2 * np.pi), label=name)
    ax.set_xlabel("parameter")
    ax.set_ylabel("heading / $2\\pi$")
    ax.legend()



.. image-sg:: /api/gallery/topology/curves/images/sphx_glr_plot_02_hopf_umlaufsatz_002.png
   :alt: plot 02 hopf umlaufsatz
   :srcset: /api/gallery/topology/curves/images/sphx_glr_plot_02_hopf_umlaufsatz_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1174d2660>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.091 seconds)


.. _sphx_glr_download_api_gallery_topology_curves_plot_02_hopf_umlaufsatz.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/curves/plot_02_hopf_umlaufsatz.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_hopf_umlaufsatz.ipynb <plot_02_hopf_umlaufsatz.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_hopf_umlaufsatz.py <plot_02_hopf_umlaufsatz.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_hopf_umlaufsatz.zip <plot_02_hopf_umlaufsatz.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_